1. Derive (i.e. prove) Boole's rule by making use of a Lagrange polynomial of order 4. You may use similar reasoning as was outlined in Week 6 for deriving Simpson's rule.

2. Mathews and Fink show how Simpson's rule can be used to approximate the solution of an integral equation (Problem 7, page 377). The procedure is outlined as follows:

Let the integral equation be given as v (x) = x² + 0.1₀∫¹(x² + t)v (t) dt. This could, for example, be the expression for the velocity of some object at position x. Note here that t is just a dummy variable used for integration purposes. To solve this integral equation via Simpson's rule with h = 0.5, we let t₀ = 0, t₁ = 0.5 and t₁ = 1. We then have

₀∫¹(x² + t)v dt ≈ 0.5/3 [(x_n² + 0)v_o + 4 (x_n² + 0.5)v₁ + (x_n² + 1)v₂]      (1)

Substituting Eq (1) into the integral equation we then have

v(x_n) ≈ x²_n + 0.1 {1/6[(x²_n + 0) v₀ + 4 (x_n² + 0.5. v₁ + (x_n² + 1) v₂]}    (2)

Substituting x₀ = 0, x₁ = 0.5, x₂ = 1 into Eq (2) gives us a system of linear equations:

v₀ = 0 + 1/60 (0 × v₀ + 2v₁ + v₂)

v₁ = 0.25 + 1/60 (0.25v₀ + 3v₁ + 1.25v₂)

v₂ = 1 + 1/60 (v₀ + 6v₁ + 2v₂)

which can be solved to give v₀ = 0.0273, v₁ = 0.2866, v₂ = 1.0646. Substituting these values back into Eq (2) and simplifying the algebra gives us the solution to the integral equation:

v (x) ≈ 1.037305x² + 0.027297             (3)

One can check the validity of the solution by substituting it back into the right hand side of the integral equation, integrating and simplifying the right hand side.

This should compare well with the approximate solution given by Eq (3). This is a technique known as a self-consistency check and is common throughout applied mathematics, science and engineering.

(a) Using the ideas presented above, use the 3/8 Simpson rule with h = 0.5 to approximate the solution of the integral equation given by

v (x) = x³ + 0.25₀∫^1.5 (x³ + t)v(t) dt       (4)

Make sure you use any relevant algorithms at your disposal to solve the system of linear equations required to obtain the required v_i coefficients.

(b) Do a self-consistency check of the solution and comment on the accuracy of the numerical solution.

3. The so-called ‘predator-prey' model is a classic example of a non-linear system of differential equations in which the complex relationship between populations of predators and prey co-exist.

While a simplistic model, it nevertheless captures some essential information about population dynamics and has been one of the most studied systems of mathematical biology that has profoundly increased our understanding of dynamical systems and chaos. The system of coupled equations is given as

x^.(t) = α_x (t) - β_x (t) y(t)

y^.(t) = γ_x (t) y (t) - δ_y (t)                 (5)

where x represents the number of prey, y represents the number of predators, t is time and α, β, γ, δ are model input parameters. Using values of α = 4, β = 2, γ = 3, δ = 3, solve via the fourth order Runge-Kutta method the system of differential equations over the time interval [0, 10] with h = 0.01 if:

(a) x (0) = 2, y (0) = 1. Plot the number of predators and prey as a function of time on the same plot and comment on the meaning of the plot.

(b) x (0) = 2, y (0) = 20. Again plot the numbers of predators and prey and comment on the significance.

(c) For both (a) and (b) repeat the calculations with time in the time range [0, 100] and now plot y as a function of x. Comment on the structure you see in the plot.

4. Consider the third order differential equation given by

d³y/dx³ + 3d²y/dx² + xdy/dx = x²y                 (6)

given that

y(0) = 2, dy/dx|_{x = 0} = 1, d²y/dx²|x = 0 = -1,

(a) Transform the differential equation into three first order coupled dif- ferential equations, with corresponding initial conditions.

(b) Solve Eq (6) by solving the system of coupled first order differential equations obtained in part (a) using the fourth order Runge-Kutta algorithm in the range x ∈ [0, 3] and plot the solution.